Simplified Calculation of Short-Term Deflection in Prestressed Two-Way Flat Slabs

Transcription

1 ACI STRUCTURAL JOURNAL Title no. 103-S86 TECHNICAL PAPER Simplified Clcultion of Short-Term Deflection in Prestressed Two-Wy Flt Slbs by Shih-Ho Cho nd Antoine E. Nmn While the deflection control of reinforced concrete two-wy slbs is generlly chieved by stisfying code-specified minimum thickness, the ACI Building Code requires the immedite short-term deflection of prestressed concrete slbs to be computed by usul methods or formuls for elstic deflections nd compred with the llowble vlues given in the Code. The clcultion of deflections for twowy slbs, however, is complicted even if elstic behvior cn be ssumed. In this pper, simplified expressions for computing the short term elstic deflection of prestressed two-wy flt slbs re proposed; they re bsed on results from extensive finite element nlyses nd re pplicble to the computtion of live lod deflections. The proposed equtions ccount for two-wy ction, support conditions, pnel spect rtio, nd loding ptterns. Predicted mximum deflections using the proposed equtions re compred with results obtined from more exct but less nlyticlly involved procedures proposed by other reserchers nd by ACI Committee 35. It is found tht the proposed equtions hve the dvntge of being reltively simple while significntly reducing computtionl effort in comprison with other procedures. Furthermore, the predicted results re sufficiently ccurte nd on the conservtive side Keywords: deflection; prestress; slb. INTRODUCTION Prestressed concrete flt slbs hve number of fetures tht distinguish them from reinforced concrete flt slbs. First, by pproprite prestressing, they cn be designed not to crck under full service lod. Second, generlly ll the ded lod is blnced by the prestressing force. Thus, the immedite deflection due to full live lod, or portion of it, is much smller for prestressed slb thn for n equivlent reinforced concrete slb. The deflection of flt slb (with rectngulr pnels supported by columns t four corners only) cn be clculted using the elstic plte eqution proposed by Timoshenko nd Woinowsky-Krieger (196); it is bsed on the following ssumptions: The cross-sectionl dimensions of the columns re smll compred to the pnel dimension nd cn be neglected; nd The slb hs n infinite number of pnels in both directions. The mximum deflection ws obtined ssuming lternte strip loding on the slb in one direction only, such s in continuous bems. Assuming the long spn to short spn rtio (l / ) is equl to 1, tht is, squre pnel, then in which wl D (1) D where equls the mximum deflection, w equls the trnsverse lod per unit re, l equls the long spn of the rectngulr pnel from column center to center, equls the short spn of the rectngulr pnel, equls the modulus of elsticity of concrete, D equls the flexurl rigidity per unit width, h equls the slb thickness, nd ν equls the Poisson s rtio. Using ν 0.2 for concrete (ν 0.15 ~ 0.25 for concrete [ACI Committee ]), the deflection of squre pnesed on Eq. (1) nd (2) is reduced to For slb subjected to checkerbord loding (lternte strip loding in two directions), the deflection in the center of squre pnel is ( 1 ν 2 ) It is evident from Eq. (3) nd () tht the loding pttern hs mjor influence on the mgnitude of deflection. Additionlly, the deflection depends on the rtio of long spn to short spn (Timoshenko nd Woinowsky-Krieger 196). In slb with finite dimensions where the dimensions of columns cnnot be neglected, however, Eq. (1) my not led to the mximum deflection, nd the mximum deflection my occur under the checkerbord loding cse. In generl, the mximum deflection in flt slb cn be expressed in the following form wl wl k wl where k equls the coefficient, depending on the rtio of long spn l to short spn of the pnel, loding ptterns, nd ACI Structurl Journl, V. 103, No. 6, November-December MS No received December 22, 2005, nd reviewed under Institute publiction policies. Copyright 2006, Americn Concrete Institute. All rights reserved, including the mking of copies unless permission is obtined from the copyright proprietors. Pertinent discussion including uthor s closure, if ny, wile published in the September- October 2007 ACI Structurl Journl if the discussion is received by My 1, (2) (3) () (5) 850 ACI Structurl Journl/November-December 2006

2 ACI member Shih-Ho Cho is Post-Doctorl Reserch Fellow in the Deprtment of Civil nd Environmentl Engineering t the University of Michign, Ann Arbor, Mich., where he lso received his PhD in structurl nd mterils engineering. He is member of ACI Committee 08, Bond nd Development of Reinforcement, nd Joint ACI-ASCE Committee 352, Joints nd Connections in Monolithic Concrete Structures. His reserch interests include prestressed concrete nd the seismic behvior of highperformnce fiber-reinforced cementitious composites. Antoine E. Nmn, FACI, is Professor in the Deprtment of Civil nd Environmentl Engineering t the University of Michign. He is member of ACI Committees 363, High-Strength Concrete; 0, Fiber Reinforced Polymer Reinforcement; 5, Fiber Reinforced Concrete; 59, Thin Reinforced Cementitious Products nd Ferrocement; nd Joint ACI-ASCE Committees 33, Concrete Bridge Design, nd 23, Prestressed Concrete. His reserch interests include high-performnce fiber-reinforced cement composites nd prestressed concrete. support conditions (ssuming the Poisson s rtio is constnt). Tests on prestressed concrete flt slbs with different boundry conditions show tht k vlues rnge from (Nwy nd Chkrbrti 1976) to 0.15 (Scordelis et l. 1959). Bsed on these experimentl results s well s the elstic plte theory, Nmn (200) proposed n pproximtion eqution to clculte the short-term deflection l wl for rectngulr pnels in which l It is noted tht Eq. (6) is essentilly bsed on the theory derived by Timoshenko nd Woinowsky-Krieger (196), in which the plte slope t column fce ws ssumed to be zero, implying rigid support condition. The clssic elstic plte theory tkes into ccount the twowy ction of slbs nd is precise procedure. It requires considerble mthemticl effort, however, nd is consequently not prcticl for routine design. ACI Committee 35 (1991) proposes n lterntive pproch, tht is, the crossing bem method, bsed on simple bem deflection equtions. In this method, the slb is divided into column strips nd middle strips in ech of the two orthogonl directions nd both strips re treted s continuous bems nd nlyzed for moments by either the direct-design method or the equivlent-frme Fig. 1 Study exmple for prestressed two-wy flt slb system (Nmn 200). ACI Structurl Journl/November-December 2006 (6) method. Moments re then used to clculte the column strip nd middle strip deflections. Deflection of the midpnel is then obtined by superimposing the column strip deflection in one direction nd the middle strip deflection in the orthogonl direction. Bsed on deflection control provisions in the Austrlin Stndrd for Concrete Structures, Gilbert (1989) presented the following modified expression to estimte the spn-todepth rtio of post-tensioned slb systems l ( --- K l )1000E c h + w u 3w us where w u is the unblnced service lod (kp) nd w us is the sustined prt of the unblnced lod (kp). The term 3w us in Eq. (7) ccounts for the long-term deflection. K is slb system fctor tht ccounts for the support conditions of the slb pnel, the rtio of long spn to short spn of the pnel l /, nd the torsionl stiffness of the slb. By rerrnging the terms in Eq. (7), one cn find tht it hs exctly the sme form s Eq. (5). The K vlues given by Gilbert (1989) for n edgesupport two-wy slb re dependent on the l / rtio, wheres K vlues for flt slb (without supporting bems) re independent on the l / rtio. Elstic plte theory shows, however, tht deflection in two-wy flt slb is dependent on the l / rtio. Moreover, both the ACI Committee 35 (1991) nd Gilbert s (1989) pproches ssume uniform loding nd do not ccount for pttern loding, which could result in lrger deflections. While the deflection control of reinforced concrete twowy slbs is chieved by stisfying code-specified minimum thickness, the ACI Building Code (ACI Committee ) requires tht the immedite deflection of prestressed concrete slbs due to live lod be computed by usul methods or formuls for elstic deflections (ACI , Section 9.5.) nd compred with the permissible vlues (ACI , Tble 9.5(b)). Unfortuntely, the clcultion of deflections for two-wy continuous slbs is complicted nd time consuming, even if elstic behvior cn be ssumed. In this pper, simplified expression in the formt of Eq. (5) is proposed for computing the immedite elstic deflection of prestressed two-wy flt slbs; it is bsed on results from extensive finite element nlysis. This nlysis ws crried out using typicl prestressed flt slb system (Fig. 1 [Nmn 200]) nd tking into considertion three prmeters: 1. Vrious long spn l to short spn rtio of the pnel; 2. Two loding conditions, nmely, checkerbord loding nd lternte strip loding; nd 3. Two support conditions in between slbs nd columns, nmely, hinge support nd rigid support. RESEARCH SIGNIFICANCE The clcultion of live lod deflections for prestressed two-wy slbs is complicted when using current nlyticl procedures. Simplified expressions bsed on extensive finite element nlyses re proposed in this pper. It is shown tht, while computtionl effort is significntly reduced by using the proposed equtions, the predicted results re sufficiently ccurte when compred to other procedures nd re on the conservtive side. (7) 851

3 FINITE ELEMENT ANALYSIS Slb model Consider the prestressed concrete flt floor system with squre pnels shown in Fig. 1. It is proposed for n office building nd uses unbonded tendons. The columns re 91 mm (30 ft) prt, center to center in ech direction. Their cross sections re given in Fig. 1. Prestressing forces re pplied in both directions of the slb; it is ssumed tht no deflection is induced under ded lod becuse of lod blncing. In other words, ll of the ded lod is blnced by the prestressing forces. A design live lod of.79 kp (100 lb/ft 2 ) is specified. The slb thickness is mm (9 in.); 29,511 MP ( psi); Poisson s rtio ν 0.2. The spect rtio of l / is equl to 1. The entire flt slb ws sub-meshed into 3072 qudrilterl plte elements, tht is, x x mm (22.5 x 22.5 x 9 in.) for ech element when conducting the finite element nlysis. Prmetric studies with rnge of element sizes showed tht convergence ws chieved by using the selected element size. The plte element used is four-node qudrilterl plte/shell element using the Mindlin-Reissner plte theory. Ech node hs five degrees of freedom (DOF); tht is, three trnsltions nd the two in-plne rottions. The drilling DOF Fig. 2 Four-node qudrilterl plte element nd corresponding nodl DOF (Cook et l. 2002). (rottion bout the out-of-plne xis) is not included. A typicl plte element nd its corresponding nodl DOF re shown in Fig. 2 (Cook et l. 2002). To simulte the most criticl sitution, two loding ptterns were considered in the nlysis, s shown in Fig. 3. Figure 3() shows the lternte strip loding plced long the W-E direction nd Fig. 3(b) shows the checkerbord loding. One cn lso plce the lternte strip loding in the N-S direction; however, nlysis results reveled tht, in either cse, the mximum deflection is lmost identicl. Incresing the l / rtio ws chieved by incresing the number of columns in the W-E direction while keeping the element size the sme (571.5 x x mm [22.5 x 22.5 x 9 in.]). An exmple is shown in Fig., where the spect rtio of n individul pnel is incresed from 1 to 2 by dding 16 columns long the W-E direction (this is, s if dding wll long the W-E direction). Further, everything else being equl, the checkerbord type loding cn be configured either on individul pnels (supported by four columns), s shown in Fig. (), or remin the sme for ll the spect rtios, s shown in Fig. 3(b) nd (b). Results of the finite element nlysis indicted tht the ltter cse gives lrger deflection vlues. Accordingly, in this study, the checkerbord loding ws pplied s in Fig. 3(b) nd ws not vried or chnged with the spect rtios. Two support conditions, tht is, hinge support nd rigid support, were lso considered, s shown in Fig. 5. The former cse llows for rottions t the slb-to-column connections. This could occur, for instnce, in lift-slb construction due to the flexibility of lifting collrs (Nilson 1987; Zllen nd Perz 200). The cse of rigid supports restricts rottions t the slb-to-column connections, which is pplicble to conventionl cst-in-plce slb system. The ctul rottion, however, my be somewht in between. As observed by Gilbert et l. (1985), crcking in the immedite vicinity of the column of post-tensioned flt slb could result in loss of fixity, which, in turn, ffects the deflection. Anlysis results nd design recommendtions The finite element nlysis ws crried out by threedimensionl structurl progrm (RISA Technologies 2001). The mximum observed vlue of deflection for flt slb with vrious loding ptterns nd support conditions re shown in Fig. 6. Mjor findings for slbs with hinge supports re: 1) the lternte strip loding governs in ll the spect Fig. 3 Two loding ptterns for live lod (shded res): () lternte strip loding; nd (b) checkerbord loding. Fig. Two types of lyout for checkerbord loding of flt slb where l / rtio equls 2. Fig. 5 Two different support conditions for flt slbcolumn connection (W-E direction view): () hinge support; nd (b) rigid support. 852 ACI Structurl Journl/November-December 2006

4 rtios; 2) for both the checkerbord loding nd lternte strip loding cses, the mximum deflection decreses to 75% of its vlue when the spect rtio increses from 1 to 1.5; nd 3) the mximum deflection rpidly converges to n symptotic vlue when the spect rtio is lrger thn 1.5. It ws further noted tht the loction of mximum deflection is not necessrily t the center of pnel. Selected results of the mximum deflections nd their loctions for slb with hinge-supported columns re illustrted in Fig. 7. The vlues of mximum deflections for ech cse re shown nd loctions of mximum deflections re mrked. The reson tht the mximum deflection does not occur t the center of pnel cn be esily understood by nlyzing the deflections in continuous bem (such relevnt vlues cn be found in Tble 5-17 of the AISC mnul [2001]). It cn be observed tht the loction of mximum deflection in three- or fourspn continuous bem depends on the number of loded spns nd the lod ptterns, nd is not in the middle of ny spn, s one my intuitively guess. In the cse of flt slb with rigid supports, it is found tht the mximum deflection from checkerbord loding nd tht from lternte strip loding hve only minor differences. The mximum deflection decreses to pproximtely 60% of its vlue when the spect rtio increses from 1 to 1.5. Similr to the cse of hinge support, the mximum deflection rpidly converges to n symptotic vlue when the spect rtio is lrger thn 1.5. The mximum deflections in slbs with either rigid or hinge support conditions nd spect rtio l / equl to 1, 2, or infinity re summrized in Tble 1 nd 2. Becuse the deflection converges to its mximum vlue t n spect rtio of pproximtely 1.5, the deflection vlue for l / 16 ws ssumed to be the sme s when l / equls infinity (refer to Fig. 6). Note in prticulr tht when l / is close to infinity, the two-wy slb cts s one-wy slb; in such cse, the mximum deflection cn be predicted using elstic bem theory. This cn be confirmed by compring the deflection vlues obtined from the finite element nlysis with those from bem nlysis. Good greement is shown in Tble 1 nd 2. Bsed on the nlyticl results, two equtions re proposed for computing the mximum deflection in prestressed two-wy flt slb system, s shown in Fig. 8 nd 9. Note for prcticl purposes only the results for slbs hving spect rtios from 1 to 2 re shown in the figures. Slb with hinge supports l wl wl -- l l (8) Fig. 6 Anlysis results for flt slb with vrious loding ptterns nd support conditions. Fig. 7 Selected results of mximum deflections nd their loctions for slb with hinge-supported columns. Tble 1 Mximum slb deflection with hinge supports Aspect rtio l / 1 (two-wy ction) l / 2 (two-wy ction) l / (one-wy ction) Limit (Bem theory) (one-wy ction) Lod pttern Checker Alternte Checker Alternte Checker Alternte Alternte Deflection, (0.62) 17.0 (0.67) (0.) 12.2 (0.8) 10.7 (0.2) Tble 2 Mximum slb deflection with ridge supports 11.9 (0.7) 11.3 (0.5) Aspect rtio l / 1 (two-wy ction) l / 2 (two-wy ction) l / (one-wy ction) Limit (Bem theory) (one-wy ction) Lod pttern Checker Alternte Checker Alternte Checker Alternte Alternte Deflection, (0.) (0.7) 6.3 (0.25) 6.58 (0.26) 3.30 (0.13) 3.28 (0.13) 2.97 (0.12) ACI Structurl Journl/November-December

5 Slb with rigid supports l --- l wl -- wl -- l l (9) Fig. 8 Anlysis results for flt slb with hinge support; mximum deflections obtined from two loding ptterns nd vrious spect rtios up to 2 re shown nd compred with proposed equtions. Fig. 9 Anlysis results for flt slb with rigid support; mximum deflections obtined from two loding ptterns nd vrious spect rtios up to 2 re shown nd compred with proposed equtions. Fig. 10 Coefficient k for proposed equtions. Eqution (8) nd (9) hve essentilly the sme fundmentl formt of Eq. (5), nd they show good greement with the results from finite element nlyses, s indicted in Fig. 8 nd 9. It is noted tht the proposed Eq. (8) nd (9) were first obtined by using the lest squre method for the liner expression (b m l / ) where b is the intercept nd m is the slope, ssuming the line obtined is multiplied by wl /. Then the line ws shifted up (with the sme slope) beyond ll the dt points to hve slightly conservtive prediction. Also shown in the figures, for comprison, re the vlues predicted by Eq. (6) given by Nmn (200). It is observed tht Eq. (6) underestimtes deflection vlues in the cse of hinge support condition (Fig. 8), but generlly gives good prediction for slbs with rigid supports (Fig. 9). The uthors recommend the use of Eq. (8) only for slbs constructed by the lift-slb method nd Eq. (9) to ll the other cses. FURTHER EXPLANATION OF PROPOSED EXPRESSIONS In prctice, most slb pnels hve spect rtio vlues less thn two. Vlues of the multiplier k (refer to Eq. (5), (8), nd (9)) with spect rtios less thn two for both support conditions re plotted versus l / in Fig. 10. Moreover, severl results from vrious experimentl nd nlyticl studies (Gilbert 1989; Nwy 2003; Nwy nd Chkrbrti 1976; Nilson 1987; Scordelis et l. 1959) re shown in Fig. 10, to compre with the mximum deflections predicted from Eq. (6), (8), nd (9). Mximum deflections tken from these studies re either clculted bsed on widely ccepted nlyticl procedures (such s the equivlent frme method) or obtined from experimentl results. To clculte the multiplier k, ll the mximum deflection vlues were trnslted into the formt of Eq. (5); the vlues of multipliers k obtined re plotted in Fig. 10. Detils cn be found in the Appendix. It is generlly observed tht the deflections from these studies re slightly smller thn those computed bsed on the proposed equtions; this my be likely due to the fct tht these studies did not consider pttern loding (lternte strip loding nd checkerbord loding) nd only considered uniform loding. In conclusion, Eq. (8) cn be tken s the upper bound of mximum deflection for slb with hinged supports, while Eq. (9) serves s the upper bound of mximum deflection for slb with rigid supports. In ny cse, the true vlues of mximum deflection might fll in between these two bounds becuse the support conditions modeled represent two extreme cses. Generlly, the short term deflection due to live lod is of interest in design. Whether only the ded weight of the slb is fully blnced by prestressing, or the ded weight plus portion of the live lod, the immedite deflection due to the nonblnced portion of live lod is needed. In ny cse, Eq. (8) nd (9) cn be used to conservtively clculte such deflection due to unblnced ded lod, s hs been done in Exmple 2 of the Appendix. 85 ACI Structurl Journl/November-December 2006

6 The proposed expressions re not intended for the determintion of slb thickness prior to design but for checking the deflection limit t the finl stge of the design. Usully, slb thickness proportioning is bsed on predetermined slb-to-depth rtios, s well s corrosion nd fire endurnce requirements. The punching sher resistnce cn lso be criticl for determining the thickness (Nmn 200). For instnce, the exmple slb (refer to Fig. 1) hs predetermined thickness equl to mm (9 in.). Then the immedite deflection due to live lod cn be esily predicted from Eq. (9), ssuming rigid support condition. The estimted deflection vlue is 12. mm (0.9 in.), which is much smller thn the mximum permissible immedite deflection due to live lod in the ACI Building Code (l/ mm) (ACI Committee ). This exmple lso indictes tht the proposed expressions re not over-conservtive, even though the worst condition is considered (pttern loding). In other words, it is generlly unlikely tht the proposed expression would led to thicker slbs. It is noted tht, for the cse of rigid supports, the predicted deflections is sensitive to the size of the column footprint. For instnce, nlysis of the study flt slb described herein showed tht the deflection vlue cn be reduced up to 30% if the mximum column size is doubled (ssuming l / 1, nd lternte loding condition). It cn be then concluded tht the prediction equtions suggested become conservtive for column sizes lrger thn pproximtely 10% of the spn length. SUMMARY AND CONCLUSIONS The current ACI Building Code (2005) suggests tht the immedite deflection of prestressed concrete slbs due to live lod be computed from usul nlysis, such s equivlent frme method, nd then compred with the permissible vlues. Computtion of deflection, however, cn be usully complicted nd time-consuming. The purpose of this study ws to seek simple expression for clculting the mximum short term deflection of flt slb due to its live lod bsed on comprehensive finite element nlysis. The proposed equtions ccount for two-wy ction, support conditions (hinged nd rigid supports), spect rtio of pnel, nd loding ptterns (checkerbord nd lternte strip lodings). It should be mentioned tht most of the vilble procedures to predict deflection in slbs do not ccount for loding ptterns nd thus would generlly underestimte the short term deflection. Predicted mximum deflections using the proposed equtions were compred to results obtined from other elborte nlyticl procedures or experimentl results. It ws observed tht the proposed equtions hve the dvntge of being reltively simple nd ccurte, while significntly reducing the computtionl effort in comprison with other procedures, such s the equivlent frme method. In ddition, the predicted deflections using the proposed equtions re on the conservtive side, providing usefuounds. Finlly, it should be noted tht the proposed method, while prcticl, does not ccount for ll possible prmeters, such s the effect of drop pnels, column cpitls, nd interior or edge bems. In presence of such elements, the proposed equtions my be conservtive, yet useful for rough estimtes. ACKNOWLEDGMENTS This reserch ws supported in prt by grnts from the Ntionl Science Foundtion (CMS nd CMS ), nd by the University of Michign. Any opinions, findings, nd conclusions expressed in this pper re those of the uthors, nd do not necessrily reflect the views of the sponsors. NOTATION D flexurl rigidity per unit width modulus of elsticity of concrete h slb thickness k, K coefficient depending on rtio of long spn l to short spn of pnel, loding ptterns, nd support conditions l long spn of rectngulr pnel, from column center to center short spn of rectngulr, from column center to center w trnsverse lod per unit re w L live lod w u unblnced service lod w us sustined prt of unblnced lod mximum deflection of slb ν Poisson s rtio REFERENCES ACI Committee 318, 2005, Building Code Requirements for Structurl Concrete (ACI ) nd Commentry (318R-05), Americn Concrete Institute, Frmington Hills, Mich., 30 pp. ACI Committee 35, 1991, Report on Control of Two-Wy Slb Deflections (ACI 35.9R-91), Americn Concrete Institute, Frmington Hills, Mich., 1 pp. AISC, 2001, Lod nd Resistnce Fctor Design Specifiction for Structurl Steel Buildings, 3rd Edition, Americn Institute of Steel Construction, Chicgo, Ill., pp Cook, R. D.; Mlkus, D. S.; Plesh, M. E.; nd Witt, R. J., 2002, Concepts nd Applictions of Finite Element Anlysis, th Edition, John Wiley & Sons, Inc., 78 pp. Gilbert, R. I., 1989, Determintion of Slb Thickness in Suspended Post- Tensioned Floor Systems, ACI Structurl Journl, V. 86, No. 5, Sept.- Oct., pp Gilbert, S. G.; Clelnd, D. J.; nd Long, A. E., 1985, Study of Service Lod Deflections nd Ultimte Lod Deflections of Flt Slb Structures, Deflections of Concrete Structures, SP-86, Americn Concrete Institute, Frmington Hills, Mich., 39 pp. Nmn, A. E., 200, Prestressed Concrete Anlysis nd Design: Fundmentls, 2nd Edition, Techno Press, 1072 pp. Nwy, E. G., 2003, Prestressed Concrete: A Fundmentl Approch, th Edition, Prentice Hll, 950 pp. Nwy, E. G., nd Chkrbrti, P., 1976, Deflection of Prestressed Concrete Flt Pltes, PCI Journl, V. 21, No. 2, pp Nilson, A. H., 1987, Design of Prestressed Concrete, 2nd Edition, John Wiley & Sons, 608 pp. RISA Technologies, 2001, Version.5 User s Guide, Foothill Rnch, Clif., pp Scordelis, A. C.; Lin, T. Y.; nd Ity, R., 1959, Behvior of Continuous Slb Prestressed in Two Directions, ACI JOURNAL, Proceedings V. 56, No. 12, Dec. pp Timoshenko, S. P., nd Woinowsky-Krieger, S., 196, Theory of Pltes nd Shells, 2nd Edition, McGrw-Hill Publishing Co., New York, 568 pp. Zllen, R. M., nd Perz, D. B., 200, Engineering Considertions for Lift- Slb Construction, Americn Society of Civil Engineers, Reston, V., 75 pp. APPENDIX VERIFICATION OF PROPOSED EQUATIONS Prestressed flt slb with restrined rottion Exmple 1 (Gilbert 1989) A post-tensioned, two-wy flt slb floor system is considered for n office with the following prmeters: longer spn length l 10 m (32.81 ft); shorter spn length 8 m (26.25 ft), thus l / 1.25; 28,000 MP ( psi); slb thickness h 155 mm (6.08 in.) Assuming tht the self-weight of the slb is blnced, the slb only supports 3 kp (62.7 lb/ft 2 ) live lod nd 1 kp (20.9 lb/ft 2 ) sustined ded lod. This totl lod, tht is, kp (83.6 lb/ft 2 ), is conservtively pplied in lternte strip nd the mximum deflection clculted. Tble A summrizes the results obtined nd shows tht the numericl vlues obtined from the proposed expressions re conservtive nd the clculted mximum deflection ssuming rigid support condition is close to the vlue from Gilbert s eqution. Exmple 2 (Nwy 2003) A post-tensioned, unbonded two-wy flt slb floor system for n prtment complex is ACI Structurl Journl/November-December

7 Tble A Comprison of clculted mximum slb deflections for Exmple 1 (with rottion restrint) l / 1.25 Mximum deflection, Note Eqution (8) 63.1 (2.8) Hinge support Eqution (9) 38.9 (1.53) Gilbert (1989) 39.1 (1.5) Nmn (200) 37.3 (1.7) Tble B Comprison of clculted mximum slb deflections for Exmple 2 (with rottion restrint) l / 1.1 Mximum deflection, Note Eqution (8) 6.1 (0.25) Hinge support Eqution (9).15 (0.16) Nwy (2003) 2. (0.096) Equivlent frme method Nmn (200) 3.77 (0.15) Tble C Comprison of clculted mximum slb deflections for Exmple 3 (with rottion restrint) l / 1.0 Mximum deflection, Note Eqution (8) 1.1 (0.056) Hinge support Eqution (9) 0.97 (0.038) Nwy nd Chkrbrti (1976) 0.78 (0.031) Experimentl result Nmn (200) 0.82 (0.032) Tble D Comprison of clculted mximum slb deflections for Exmple (without rottion restrint) l / 1.0 Mximum deflection, Note Eqution (8) 9.83 (0.387) Hinge support Eqution (9) 6.71 (0.26) Nilson (1987) 7.82 (0.308) Equivlent frme method Hinge support Nmn (200) 5.68 (0.22) considered with the following prmeters: the end-pnel centerline dimensions re 5.33 x 6.10 m (17.5 x 20 ft) nd the dimensions for the interior pnel centerline re 7.32 x 6.10 m (2 x 20 ft); 27,790 MP ( psi); slb thickness h 165 mm (6.5 in.); nd live lod 1.92 kp (0 lb/ft 2 ). The totl ded lod is.8 kp (101 lb/ft 2 ) nd only 70% the ded lod is blnced, tht is, 3.39 kp (70 lb/ft 2 ), thus the slb supports the live lod nd unblnced ded lod equl to 3. kp (71 lb/ft 2 ). Using w w L + w u 3. kp nd clculting the mximum short-term deflection of the exterior pnels (l / 1.1) led to results shown in Tble B. It cn be observed tht the clculted vlues bsed on the proposed eqution (Eq. (9)) re slightly higher thn the vlue bsed on equivlent frme pproch s obtined by Nwy. This is due to the fct tht the unblnced lod in this study is pplied using either checkerbord loding pttern or lternte strip loding pttern rther thn the uniformly distributed pttern used by Nwy. Exmple 3 (Nwy nd Chkrbrti 1976) Nwy nd Chkrbrti crried out experiments on four post-tensioned flt pltes with different distributions of prestressing steel. Except for the steel lyout, ll other prmeters such s totl mount of prestressing steel, boundry conditions, effective prestressing stresses, nd mteril properties were the sme. The plte hd four pnels with centerline dimension of Tble E Comprison of clculted mximum slb deflections for Exmple 5 (without rottion restrint) l / 1.0 Mximum deflection Note Eqution (8) Eqution (9) Scordelis et l. (1959) Nmn (200) 0.19 wl 0.13 wl 0.15 wl 0.11 wl Hinge support Experimentl result Hinge support x m (30 x 30 in.) for ech pnel (l / 1.0). The center column hd dimension of x mm ( x in.) nd provided rottion restrint to the pltes. 29,670 MP ( psi); slb thickness h 33.3 mm (1.31 in.) Uniform loding w 2. kp (50 lb/ft 2 ) ws pplied on ll pnels through wter-filled pressure bg. Experimentl results showed tht, even if the steel lyout were vrious, ll specimens hd lmost identicl mximum deflections. Averged vlue is shown in Tble C, s well s estimtes from Eq. (6), (8), nd (9). The prediction bsed on Eq. (9) is close to the test dt, but slightly lrger becuse no pttern loding ws pplied during the tests. Also, Eq. (6) showed good greement with the test result. Prestressed flt slb without rottion restrint Exmple (Nilson 1987) A prestressed flt plte lift slb is supported by steel columns spced 7.62 m (25 ft) on centers in ech direction. Lods wile trnsferred from the slb to the columns by squre lifting collrs; however, the collr detil trnsmits no significnt bending moment to the columns. This implies tht the slb-to-column connection is free to rotte. 27,790 MP ( psi); slb thickness h 178 mm (7 in.); live lod is 2. kp (50 lb/ft 2 ). The selfweight of the slb nd the dditionl ded lod re blnced by the prestressing. The clcultion of mximum short-term deflection of the typicl pnels (l / 1.0) is crried out nd the results re given in Tble D. They indicte tht Eq. (8) is more suitble for this cse becuse it ccounts for the possible rottions (hinge support). As lso due to the effect of pttern loding, the clculted vlue (Eq. (8)) is slightly higher thn tht bsed from the equivlent frme method. Exmple 5 (Scordelis et l. 1959) Scordelis et l. tested four-pnel prestressed flt slb tht hd centerline dimension of 2.13 x 2.13 m (7 x 7 ft) for ech pnel (l / 1.0). A rocker nd roller rrngement t ech support llowed rottions to occur. Uniform loding ws pplied on the slb nd the mximum deflection ws expressed using the form of Eq. (5). As cn be seen in Tble E, the vlue suggested by Scordelis et l. (1959) flls in between the vlues obtined from Eq. (8) nd (9). In this cse, Eq. (8) is more suitble becuse it tkes into ccount the possible rottion t supports. It cn be concluded from Exmples 1, 2, nd 3 tht, in generl, Eq. (9) is more suitble for the flt slb system in which slb rottion t support is restrined. Further, bsed on Exmples nd 5, Eq. (8) is more suitble for the flt slb system in which slb rottion t support is llowed. Results bsed on Eq. (6) (Nmn 200) re lso shown for comprison purposes. In generl, however, the clculted vlue from Eq. (6) tends to underestimte the deflection when the rottion t slbto-column connection is llowed. 856 ACI Structurl Journl/November-December 2006